basins of attraction of the two-dimensional “poor man’s …acfd/ijbc.pdf · 2006-01-03 ·...

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Basins of Attraction of the Two-Dimensional “Poor Man’s Navier–Stokes Equation” ABSTRACT

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Page 1: Basins of Attraction of the Two-Dimensional “Poor Man’s …acfd/ijbc.pdf · 2006-01-03 · Basins of Attraction of the Two-Dimensional “Poor Man’s Navier–Stokes Equation”

Basins of Attraction of the Two-Dimensional“Poor Man’s Navier–Stokes Equation”

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ABSTRACT

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Page 2: Basins of Attraction of the Two-Dimensional “Poor Man’s …acfd/ijbc.pdf · 2006-01-03 · Basins of Attraction of the Two-Dimensional “Poor Man’s Navier–Stokes Equation”

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Page 7: Basins of Attraction of the Two-Dimensional “Poor Man’s …acfd/ijbc.pdf · 2006-01-03 · Basins of Attraction of the Two-Dimensional “Poor Man’s Navier–Stokes Equation”

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13 Divergent

12 Broadband w/o fundamental

11 Broadband w/ different fundamental

10 Broadband w/ fundamental

09 Noisy quasiperiodic w/o fundamental

08 Noisy quasiperiodic w/ fundamental

07 Noisy phase lock

06 Noisy subharmonic

05 Quasiperiodic

04 Phase lock

03 Subharmonic

02 Periodic w/ different fundamental

01 Periodic

00 Steady

(e)

1.0 2.0 3.0 4.01.5 2.5 3.5

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−0.6

−0.2

0.2

1.0 2.0 3.0 4.01.5 2.5 3.5

−0.8

0.0

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0.4

0.8

1.2

1.6

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1.0 2.0 3.0 4.01.5 2.5 3.5

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0.5

0.0

2.5

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3.0

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1.5

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0.5

0.0

2.5

2.0

1.0

3.0

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β

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Page 9: Basins of Attraction of the Two-Dimensional “Poor Man’s …acfd/ijbc.pdf · 2006-01-03 · Basins of Attraction of the Two-Dimensional “Poor Man’s Navier–Stokes Equation”

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100 X 100

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a (0) a (0)

a (0)a (0)

b(0)

50 X 50

b(0)

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` j,��_ afn�aJk,���dj,p n�Z|��_ af^�s 8 � 8R� ` _ �}r9j!n�_��[�m_ �<��_�r,b ����^D�0�|k#������s3r�b�nV_ �<Z|^JkI_ �|k!` jI��_ afn�a�k,����j,p�n�Zm��_ a^�s 8 � 88R� 2 `�_ ��r,j|� 1 k,^�� ^J jI�|kvz�nBr9��^�_ j0^Vs/r,b �± n�j,��_ j�n�j0a�k,���fkI�mª�_ �m�w�m_Rr��j�p��|kvsc^Jk,�h�|�?�0�±Zm^J_ Zm��` � �r9b�nVr0r9b ��j0p n�kIr,��Z|`���n�k��n�jI��_��wn�p�n�p p��]n�kIjr9^¯� ��jIp ��n]ywn{sck,n�Zmr9n���x#r9^Jp�^J��^�a�y2~�n��}r9b ^�` aJbw�0�>b�n]z��>_ ^�rnVr,r,�|�hp�r9�m�Mnfk,��aJ^JkI^J` j�p kI^�^VsK^�s6r9b���j|�

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3.3 Case 3 (subharmonic / noisy subharmonic): F � « S 22 � L � 8 S « �¢�_ Z]nVj,�&« ��&Z|^J_Rr,��_�` �gr9^°zVn�kIy F n�_�� L j,^ r9b�nVr§�0�&Z]n�_ j,b�^�� ��©�n��hp ���mj§^�s¯r,b ���|_Rr9��k,�

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^�r9b �mk¯b n�_ ���6r9b �fzVn�k,��nVr9��^J_ j�^�s�r,b �h_2`��w�mk,��Z|n��FzVn���` �mj¯^Vs�j,�V�m�l_ �mj,jhn�_ �Yj,�mZ|^J_ �2��^JkI� �|k¯jIr9kI` Zmr,` k,�sc` _ Zmr,��^J_ � �^�r,bM�nVj,�|�M^J_[^J_ ��y�^J_ �{Z|^J�hp�^J_ �|_Rrq^�s�r9b ��zJ�m��^2Z|�}r�y ����mr��0�|�m_�r9b ��r��0^�p b�nVj,������^2Z(���m��n�j,��_ j¯n�k,�waJ�|_ �mk9n�����yor��0^o^JkI� �|kIj�^�s���n�a�_ ��r,` � �faJk,�|nVr9�mk{r,b�n�_&r9b �hz�nVk,��nBr9��^�_ j�^�s�r,b �|jI��j,^���`�r,��^J_p�n�k9nV�w��r9�|kIj|��_�` �h�|k,��Z]n��RzVn���` �|j�n���^�_ a!r9b �0Z(b�n�^�r,��Z0nVr,r,k9n�Z�r9^Jk6n�_ ��n�k,�¶r9b�` j6z2��j,�� ���V� i b���j6�w��r9b ^2�¯^Vs��_Rr9�mk,p kI�mr9��_ a¯r,b �% n�j,��_dp ��^Vr9j0n�_ ��r,�|jIr,��_ a¯p/^���_Rr9j��l��r9b���_h�]n�Z(b§�nVj,��_wsc^Jk0Zm^J_ jI��jIr,�|_ Z�yw�n�j!nV����`�� �|�r9^�p kI�mz2��^�` j,�}yJ� 1 ` kIr,b �|k0� ��jIZ|` jIj,��^J_§��_�r,b �%sc^J����^D�l��_�a¯jI�|Zmr,��^J_�j�l������p�k,^Dz2��� ��nV_d��©�p ��n�_�nVr,��^J_w^�sGr9b �p k,^Jp��|klp�nVk9n��h�mr,�|k,j�r,^f/��` jI�|����_§sc`�r9` kI��jIr9`�� ���mj

3.5 Case 5 (noisy phase lock / noisy quasiperiodic w/ fundamental): F � « S � « 22 � L � 8 S � 21�nVj,� � ��j�jI`��hZ|���|_Rr¶�mz2�����|_ Zm�r9b�nBrKsc` kvr9b �mkKjIr,` � ���|jK^Vs�r9b �0Z|^J_ jI��jvr9�|_�Zmy�^Vs�j,^J��`�r9��^J_�p n�k9n��h�mr,�|kIj|�

r9b �dZm��kIZ|` �hjIr9n�_ Z|�mj�sc^Jk��lb ��Z(b noj,p��|Zm��ª��m� p n�k9n��h�mr,�|k{�l��������dZ|^JkIk,�mZmr]��n�_��°b ^D� r,^�nDz�^J���°�h��j#���� �m_�r,��ª��m�YjI^J��`�r9��^�_g�n�j,��_ j¯n�k,�f_ �m�|� �m����¢�_YZ]n�jI�fª�zJ�w�0�hZ|^J_ jI��� �mk�^J_ ��y�r9b �f����_ ��r9��Zh�m_ �|kIa�y&n�_ ���nVr9_��|j,jqZ|^�_�r,^J` kIj|��� ��j,p ��nDy��|�[��_ 1 ��aJj|� � � nR��n�_�� � �����kI�|jIp/�mZmr9�}zJ�m��yJ�

(a)

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wer (

dB)

Frequency, ƒ, (normalized)

Scaled time

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v v

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ity

Scaled time

(e)

(d)

a (0)

a (0)

b(0)

00

00

1

1

1

1

b(0)

1 ��aJ`�k,� � ³ 1^J_Rr,^J` k,jqsc^Jk F � « S � « 22 � L � 8 S � 2 ` j,��_ a � nR�!�2��_ �mr,��Z��|_��|k,aVyon�_�� � ��l��nVr,_ �|jIj>n���^J_ a�l��r9b§r,b � � ZD��p�^D��mkqjIp/�mZmr,k9n�nV_ � � ����� � �D����© nV�wp ����Z|^J�hp�^J_ �|_Rr!r,���h��j,�|kI���mj|�

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1�nVj,� 2 � ��jIp ��n]y2j�Z(b�n�k,n�Zmr,�|kI��jIr,��Zmj Z|^J�h�w^�_>r9^!�h^JjIr n�j,��_{p���^�r,j���� �m_Rr9��ª �|����_>r,b ��j�jvr9` ��y/³±xI����_ �mj,~^�slj,^J��`�r9��^J_ j�n���^J_�aMr9b��d� ��nVaJ^J_�n���j�nV_ � n�jIy2�h�w��r9kvyJ�F��_Yr9b ��j�Z]nVj,��� � `�r?�h^JjIr�^�s3r9�m_°jIy2�h�w��r9kIy��n�/^�`�rlr9b � ; � � N � n�_ �4��j,^J�h�mr,���h�|j|� r,b � ; � � · J N � nB©��|j|�¢�_Mr9b���jlZ]n�jI�{r,b �|jI��nB©��mj � �l��r9b[j,^��w���©�Z|�mp�r9��^J_ j¶n���^J_ a ; � � ·�J N � ��n�k,�����mr9�mk,�h��_ �m��r9^�Zm^Jk,kI�|jIp/^J_���r9^�_ ^J��jIy{��`�n�jI��p��|k,��^�����Z|�}r�y��l��r,b�r9b �sc` _ ��n��h�|_Rr9n�� sck,�|��` �m_ ZmyMjIr,������p k,�mj,�m_�r � �^�r,r,^J� p ��^�r9jl^�s 1 ��aJjm� � � Z(�2�D�I��� i b���p kI�|j,�m_Rr�n�jvy��h�h�mr9kvy^�str9b��¯��nVr,_ �|jIj�Z|^J_Rr,^J` k,jm� 1 ��a � � � �������jqj,�m�|_gsc^Jkq�n�j,��_ j%^VsF_ ^J��jIy[p b�n�jI������^2Z(���|��/�mb�n]z���^Jk|��� ��jIr9��_��aJ` ��jIb �|���yhr9b��>p/^D�0�|k�jIp/�mZmr,k9n?n�_ �wr9���h�qj,�|kI���mj�^�s�r,b �%r9^�p§n�_ �d�h����� ���lp ��^�r9j0^�s 1 ��a � � � Z(�2�D��� n�_ ���j¶n�aRnV��_?� ` �lr9^�r,b �lk,�mz��|kIj9n���^�s//�mb�n]z2��^JkF^�s4��_ ����z2��� ` n��2z��|��^�Zm��r�y?Zm^J�wp�^J_ �m_Rr9j|� i b �|jI�%aJ�m_ �|k,n����}r9���mjn�k,���0^JkIr,b�yd^�st^J`�klnVr,r,�|_Rr9��^J_���n�_��Mb �mk,���0�{sc^2Z|` jl^J_§r,b ��Z]n�` j,nVr9��^J_<^�s�j,` Z(bMp/�mZ|` ����n�kI��r9���|jm�

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� <D>E@BA&C � F � <D>EC H ·QJ � <D>EC K JRL � <D>EC � <?>�C S � ·92 �i b ��jl��j!kI�|jIp/^J_�j,�������sc^Jk�r9b��{`�_ ����` ������_ �|j!^VsKjI^J��`�r,��^J_�jln���^J_�a ; � � N � nV_ �[n���jI^h��©�p ��n���_ j��lbRy<r9b �@�7 Z|^JkIk,�m��nVr,��^J_¯��jFn��}��n]y2jFp�^Jj,�}r9��z��!n���^J_ a±r,b ��jK����_����J��_ � �|p��|_�� �|_Rr¶^�s�j,^���`�r,��^J_{r�y�p���� i b��!j9n��h�r�y2p/�^�s¶n�_�nV��y2j,��j%Z]nV_o/�¯p/�mkIsc^JkI�w�m�Msc^Jklr9b�������_ ��^�s¶¢(� 1%� j�n���^J_ a ; � � · J N � ��nV_ �o^J_ �{ª�_ � j%��� �m_�r,��Z|n���|��`�nVr,��^J_ jl^�s6r9b���sc^�k,�[³

� <?>�@BA&C � � <?>�C F H ·�J � <?>�C K JML � <?>�C H ·QJ � <?>�C K � � ·E� nR�� � F J L � � <?>�C�H ·QJ � <?>�C!K,S � ·�� ��

i ^?��_Rz��|jIr,��aRnBr9�%��_dn��h^JkI�±aJ�m_ �|k,n��4�wn�_ _��|k0r,b �%^2Z|Zm` k,kI�|_ Zm��^�sGjIy2�h�w��r9kI���|j���_wr9b �±�n�j,��_d�wn�p��p ��_ a�j^�s r,b ��j0jIr9`���yw�0�>��|aJ��_��l��r,b�r9b��%aJ�m_ �|k,n��/sc^JkI��` ��n�sc^Jk�r9b �%Z|^2^Jk,� ��_�nVr,�|j^Vs ¢(� 1%� j!jIy2�h�w��r9kI��Zn�/^�`�r!r9b � ; � � N � nB©���j|³

; � A � N �! � � ·�W nR�N � A � ; �! S � ·9W ��

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